Lemma A.1 ([1]}, p.172)
For the \(\omega \) -Morse index \(\phi _\omega (x)\)  and nullity \(\nu _\omega (x)\)  of the solution \(x=x(t)\) 
and the \(\omega \) -Maslov-type index \(i_\omega (\gamma )\)  and nullity \(\nu _\omega (\gamma )\)  of the
symplectic path \({\gamma }\)  corresponding to \(x\) , for any \(\omega \in {\bf U}\)  we have \( \phi _\omega (x) = i_\omega (\gamma )\) , and \(\nu _\omega (x) = \nu _\omega (\gamma ). \)

